Question

Air pollution control specialists in southern California monitor the amount of ozone, carbon dioxide, and nitrogen dioxide in the air on an hourly basis. The hourly time series data exhibit seasonality, with the levels of pollutants showing patterns that vary over the hours in the day. On July 15, 16, and 17, the following levels of nitrogen dioxide were observed for the 12 hours from 6:00 A.M. to 6:00 P.M.

Click on the datafile logo to reference the data.

July 15:	25	28	35	50	60	60	40	35	30	25	25	20
July 16:	28	30	35	48	60	65	50	40	35	25	20	20
July 17:	35	42	45	70	72	75	60	45	40	25	25	25

Use a multiple linear regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data:
	Hour1 = 1 if the reading was made between 6:00 A.M. and 7:00A.M.; 0 otherwise
	Hour2 = 1 if the reading was made between 7:00 A.M. and 8:00 A.M.; 0 otherwise
	.
	.
	.
	Hour11 = 1 if the reading was made between 4:00 P.M. and 5:00 P.M., 0 otherwise
	Note that when the values of the 11 dummy variables are equal to 0, the observation corresponds to the 5:00 P.M. to 6:00 P.M. hour.
	If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) Do not round intermediate calculation.
	Value =  +  Hour1 +  Hour2 +  Hour3 +  Hour4 +  Hour5 +  Hour6 +  Hour7 +  Hour8 +  Hour9 +  Hour10 +  Hour11
(c)	Using the equation developed in part (b), compute estimates of the levels of nitrogen dioxide for July 18.
	If required, round your answers to three decimal places. Do not round intermediate calculation.
	6:00 a.m. - 7:00 a.m. forecast 7:00 a.m. - 8:00 a.m. forecast 8:00 a.m. - 9:00 a.m. forecast 9:00 a.m. - 10:00 a.m. forecast 10:00 a.m. - 11:00 a.m. forecast 11:00 a.m. - noon forecast noon - 1:00 p.m. forecast 1:00 p.m. - 2:00 p.m. forecast 2:00 p.m. - 3:00 p.m. forecast 3:00 p.m. - 4:00 p.m. forecast 4:00 p.m. - 5:00 p.m. forecast 5:00 p.m. - 6:00 p.m. forecast
(d)	Let t = 1 to refer to the observation in hour 1 on July 15; t = 2 to refer to the observation in hour 2 of July 15; ...; and t = 36 to refer to the observation in hour 12 of July 17. Using the dummy variables defined in part (b) and ts, develop an equation to account for seasonal effects and any linear trend in the time series.
	If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
	Value =  +  Hour1 +  Hour2 +  Hour3 +  Hour4 +  Hour5 +  Hour6 +  Hour7 +  Hour8 +  Hour9 +  Hour10 +  Hour11 +  t
(e)	Based on the seasonal effects in the data and linear trend estimated in part (d), compute estimates of the levels of nitrogen dioxide for July 18.
	If required, round your answers to three decimal places.
	6:00 a.m. - 7:00 a.m. forecast 7:00 a.m. - 8:00 a.m. forecast 8:00 a.m. - 9:00 a.m. forecast 9:00 a.m. - 10:00 a.m. forecast 10:00 a.m. - 11:00 a.m. forecast 11:00 a.m. - noon forecast noon - 1:00 p.m. forecast 1:00 p.m. - 2:00 p.m. forecast 2:00 p.m. - 3:00 p.m. forecast 3:00 p.m. - 4:00 p.m. forecast 4:00 p.m. - 5:00 p.m. forecast 5:00 p.m. - 6:00 p.m. forecast
(f)	Is the model you developed in part (b) or the model you developed in part (d) more effective?
	If required, round your answers to three decimal places.
	Model developed in part (b) Model developed in part (d) MSE
	- Select your answer -Model developed in part (b)Model developed in part (d)Item 54
	Justify your answer.

Answer 1

Step-1

Forecasting is a technique which helps in predicting the future data based on the present data or situation. It is analyzed by trend analysis. Time series is a set of observation measured at successive points in time or over successive period of time.

Step-2

a.

Construct the time series plot using XLMINER software, the procedure to make the time series plot is given as below:

1. Write down the provided data into spreadsheet, the screenshot is shown below:

2. Select the provided data range and then click on the “XLMMINER” Platform tab in the ribbon.

3. From the “Data Analysis” table select the “Explore” option. In the Explore option select the “Chart wizard” option.

A new dialog box will appear, select the “Line chart” option and press “Next” option. Now select the “level and time period” press “level” tab. Select “time period” then press “Next” and select “level” and again press “Next”. Press “Finish” option, the screenshot of the obtained time series plot is shown below:

The above time series plot indicate seasonal pattern in the level of ozone, carbon dioxide and nitrogen dioxide.

Step-3

b.

The multiple linear regression models given as:

1

Here, the intercept is the predicted value of when are equal to zero, and are the slope coefficients. According to the provided criteria introduce the dummy variables for 11 hours for three days levels. Thus the obtained dummy variables for the 11 hours will be:

Step-4

Cosider “Hour1…Hour11” as explnatory variables and “level” as dependent variable, regressing level on explanatory variable in Excel as follows:

1. Select the provided data range and then click on the “XLMINER” Platform tab in the ribbon.

2. Select the “Predict” in the “Data Mining” group and select the “Multiple Liner Regression” option.

3. A new dialog box will appear, select the explanatory variable in the “Selected variables” box. And select the dependent variable in the “Output variable” box. The screenshot is shown below:

Step-5

4. Click “Next” option in the above dialog box, a new dialog box will appear. Select the options as shown below:

5. Press “Finish” option in the above dialog box, the screenshot of the obtained regression analysis is shown below:

According to the above output, the multile regression equation for the seasonal effects is given as:

Step-6

c.

Predict estimates of the level of nitrogen dioxide for july 18, use the multile regression line obtained in the part (b). Calculat the forecast for each hour as shown below:

step-7

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Step-9

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Step-13

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Step-16

d.

Now, introduce new variable to account the seasonal effect in the data for, for this add new explanatory variable “Time (t)” in the data. The screenshot of the data file is shown below:

Step-17

Now, follow the same procedure as done in the part (b), the screenshot of the output is shown below:

According to the above output, the multile regression equation to predict the seasonal effects and liner trend is given as:

Step-18

e.

Predict hourly forcast for July 18, use the multile regression line obtained in the part (d). The forecast for the each hour for July 18 is calculated as below:

Step-19

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Step-21

f.

According to the results obtained in the part (b), the minimum MSE for the hour is calculated as

According to the results obtained in the part (d), the minimum MSE for the hour is calculated as:

Hence, the mean squared error for the model from the part (d), which include the seasonal effects and linear trend is smaller than the mean squared error for the model from the part (b), which include the seasonal effects. So, the model obtained in the part (d) will be more effective. This supports initial decisions gotten in review of the time series plot created in part (a) and the data show a linear trend with seasonality.